So I want to buy a heater and search for its model number to find
who sells it. Last week Bing Lee were showing it for
$850 before it mysteriously changed to $1,095 early this week. I
just called the store to find out the real price and it's $845.
WTF?

Why would they put a price that seems designed to turn you off
buying it on the site? If I hadn't seen the cheaper price a week ago,
I would have looked elsewhere to buy it.

A friend of mine bought a Rinnai heater and part of the reason she
bought that one was the fixed price, Apple style, with no
discounting. At least then you know what you should pay.